The probability of occurrence of two events A and B are `1//4` and `1//2` respectively . The probability of their simultaneous occurrence is `7/50`., find the probability that neither A nor B occurs .

Options

A. `25/99`
B. `39/100`
C. `61/100`
D. `17/100`

Answer

To solve the problem step by step, we will find the probability that neither event A nor event B occurs given the probabilities of A, B, and their simultaneous occurrence.

### Step-by-Step Solution

1. **Identify Given Probabilities**:
- Probability of event A, \( P(A) = \frac{1}{4} \)
- Probability of event B, \( P(B) = \frac{1}{2} \)
- Probability of both A and B occurring, \( P(A \cap B) = \frac{7}{50} \)

2. **Use the Formula for Probability of A Union B**:
The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

3. **Substitute the Values into the Formula**:
Substitute the known probabilities into the formula:
\[ P(A \cup B) = \frac{1}{4} + \frac{1}{2} - \frac{7}{50} \]

4. **Find a Common Denominator**:
The least common multiple (LCM) of the denominators (4, 2, and 50) is 100. Convert each fraction:
- \( P(A) = \frac{1}{4} = \frac{25}{100} \)
- \( P(B) = \frac{1}{2} = \frac{50}{100} \)
- \( P(A \cap B) = \frac{7}{50} = \frac{14}{100} \)

5. **Calculate \( P(A \cup B) \)**:
Now substitute these values back into the equation:
\[ P(A \cup B) = \frac{25}{100} + \frac{50}{100} - \frac{14}{100} \]
Combine the fractions:
\[ P(A \cup B) = \frac{25 + 50 - 14}{100} = \frac{61}{100} \]

6. **Find the Probability that Neither A nor B Occurs**:
The probability that neither A nor B occurs is given by:
\[ P(\text{neither A nor B}) = 1 - P(A \cup B) \]
Substitute the value we found for \( P(A \cup B) \):
\[ P(\text{neither A nor B}) = 1 - \frac{61}{100} = \frac{39}{100} \]

### Final Answer:
The probability that neither A nor B occurs is \( \frac{39}{100} \).